Free resolutions and sparse determinantal ideals

Abstract

A sparse generic matrix is a matrix whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. In this paper we extend these results by computing a minimal free resolution for all such sparse determinantal ideals. We do so by introducing a technique for pruning minimal free resolutions when a subset of the variables is set to zero. Our technique correctly computes a minimal free resolution in two cases of interest: resolutions of monomial ideals, and ideals resolved by the Eagon-Northcott Complex. As a consequence we can show that sparse determinantal ideals have a linear resolution over Z, and that the projective dimension depends only on the number of columns of the matrix that are identically zero. We show this resolution is a direct summand of an Eagon-Northcott complex. Finally, we show that all such ideals have the property that regardless of the term order chosen, the Betti numbers of the ideal and its initial ideal are the same. In particular the nonzero generators of these ideals form a universal Gröbner basis.

Lower bounds for Betti numbers of monomial ideals

“…This only works in certain cases but has the benefit that everything can be stated in terms of ideals, our subject of study. The second technique, developed in [2] works in general but relates the betti numbers of S/I to those of S/(I, x) and the module H = (I : x)/I both regarded as modules over the polynomial ring S/(x). The downside of this approach is that H need not be a cyclic module, and hence induction is not possible.…”

Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varieties

“…(3) Boocher proved in [6] that for any subgraph G of K m,n , m ≤ n, the ideal I k m (X gen G ) is radical. Combining his result with the result of Giusti and Merle, one obtains a characterization of the graphs G such that I k m (X gen G ) is prime.…”

“…Indeed, the point is that a set of m-minors of a generic matrix m×n does not generate a radical ideal in general (as it does for m = 2). For example, in the Grassmannian G (3,6) Next we look into necessary conditions for I k d (X gen G ) and I k d (X sym G ) to be prime. Lemma 7.10.…”

Universal Gröbner Bases and Cartwright–Sturmfels Ideals

“…Let σ be a term order on S. Since reg(I) ≤ reg(in σ (I)) and in σ (I) ∈ CS, we may assume without loss of generality that I is monomial. By Terai's Theorem [13,5.59] one has reg(I) = projdim(S/I * ). Since I * ∈ CS * we have projdim(S/I * ) ≤ v by Proposition 1.9 (3).…”

“…Our results might be seen as the generalization of the Bernstein-Sturmfels-Zelevinsky Theorem [4,14] (asserting that maximal minors of a matrix of variables form a universal Gröbner basis) and of the result of Sturmfels [15] and Villarreal [16] (asserting that the cycles of the complete bipartite graph give rise to the universal Gröbner basis of the ideal of 2-minors of the matrix of variables). Intermediate results in this direction have been obtained in [1,5,6,8,9,12].…”